200 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. 



Any displacement of a rigid body may be brought about in an 

 indefinite number of ways. Let three points ABC (Fig. 40) be displaced 



to A'B'C'. We may first 

 give the body a translation 

 defined by the vector A A'. 

 This will bring B to B and 

 C to C v Then through A' 

 pass an axis perpendicular 

 to the plane B^B', and 

 rotate the body about this 

 axis through the angle B^A'S 1 . 

 This brings B to B' and C 

 to a new position (7 2 . Finally 

 rotate the body about A'B' 



until C 2 arrives at C f . We have thus brought about the given dis- 

 placement by means of a succession of translations and rotations. 

 Evidently the order of these may be varied. Accordingly, 



Any displacement of a rigid body may be reduced to a succession 

 of translations and rotations. 



We have seen that a translation may be represented by a free 

 vector, a rotation, by a vector that must give the axis and the angle. 

 If we agree to draw the vector in the axis, and make its length 

 numerically equal to the angle of rotation, it will completely specify 

 the rotation, if we adopt a convention about the direction of rotation. 

 This shall be that, if the rotation is in the direction of the hands 

 of a watch, the vector shall point from face to back of the watch. 

 Vector and rotation correspond then to the translation and rotation 

 in the motion of a cork-screw, or any right-handed screw. As the 

 vector may be placed anywhere along the axis, but not out of it, 

 it has five coordinates, and may be characterized as a sliding vector. 



Translations are compounded by the law of addition of vectors. 

 The resultant of two rotations about the same axis is evidently the 

 algebraic sum of the individual rotations. The resultant of a trans- 

 lation and rotation is evidently independent of the order in which 

 they take place. 



The resultant of a rotation and a translation perpendicular to 

 its axis is equivalent to a rotation about a parallel axis, for it is 

 evident that all points move in planes perpendicular to the axis, and 

 that the motions of all such planes are alike, or the motion is 

 uniplanar. 



Now the motions of any two points in a plane determine the 

 motion of the plane parallel to itself. 



